nLab stringor bundle

Idea

In a higher version of how a spin structure induces spinor bundles, so string structure on a manifold transgresses into a kind of spin structure on its free loop space, which is “fusive” with respect to fusion of loops (along trinions). Equipped with this structure this should be a 2-vector bundle on the base space, and this is what is called the stringor bundle.

Related concepts

• spinor bundle

• fusion bundle

• Dirac operator on loop space

• What is an elliptic object?

References

The idea that String structure on a manifold is a kind of spin structure on its loop space is due to

• Edward Witten, The Index Of The Dirac Operator In Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire)

The idea and the terminology of “stringor bundles” originates with:

• Stephan Stolz, Peter Teichner, The spinor bundle on loop space (2005) [pdf, pdf]

An actual construction appears in:

• Peter Kristel, Matthias Ludewig, Konrad Waldorf, A representation of the string 2-group, [arXiv:2206.09797]

Reviewed in:

• Konrad Waldorf, The stringor bundle, talk at QFT and Cobordism, CQTS (Mar 2023) [web, video:YT]

• Matthias Ludewig, The spinor bundle on loop space and its fusion product, talk at CQTS (Apr 2023) [web, video: YT]

• Konrad Waldorf, String structures and loop spaces, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.12998]

Construction of the Connes fusion-operation on fibers stringor bundles:

• Peter Kristel, Konrad Waldorf: Connes fusion of spinors on loop space, Compositio Mathematica 160 7 (2024) 1596-1650 [arXiv:2012.08142, doi:10.1112/S0010437X24007188]

Compare also

• Alessandra Capotosti, From String structures to Spin structures on loop spaces, Ph. D. thesis, Università degli Studi Roma Tre, Rome (2016)

and other references at smooth loop space.